The aquarium salt will kill the fin rot indirectly. Bettas are very sensitive to salt, so this treatment will kill the betta. When the fin rot no longer has a host, it will die off.
http://www.bluebettausa.com/salt.htm
Fin rot (Columnaris disease) is a bacteria very resistant to salt. It would be comfortable in salinity that would instantly kill a betta.
Yes Melafix has bad side effects on bettas. Even the half strength version labeled safer for bettas has side effects. Melafix is a herbal remedy, not a medication. One difference between medications and herbals is that the side effects of herbals are rougher than the medications, while their benefits are less documentable.
http://www.fishvet.com/columnaris.htm
The first step is to clean the gravel in the betta tank and do a 50% water change. The detritus and organic build up in the water will use up the medication you add to treat the fin rot. Use a medication that contains acriflavine or metronidazole and follow the directions with it.
http://www.amazon.com/Fish-Vet-Aqua-Pro-Cure-oz/dp/B000UEENUY
A broad spectrum treatment like the Aqua Pro Cure linked above will also treat diseases that have secondarily appeared in response to the fish's weakened condition.